Solution: Set up the proportion $ \frac{7}{4} = \frac{21}{x} $. Cross-multiply: $ 7x = 84 $. Divide: $ x = 12 $. \boxed{12}

["Title: Solve Linear Proportions: A Step-by-Step Guide with $ \frac{7}{4} = \frac{21}{x} $", "Learning how to solve proportions is a fundamental skill in algebra, applicable in science, finance, and everyday decision-making. One common challenge is solving the proportion $ \frac{7}{4} = \frac{21}{x} $. In this article, we’ll walk through the solution clearly using cross-multiplication, a straightforward and powerful method—resulting in $ x = 12 $. This example illustrates not only how to solve such equations but also why understanding proportions unlocks a broad range of practical problem-solving abilities.", "## What Is a Proportion?", "A proportion states that two ratios are equal. For example, $ \frac{a}{b} = \frac{c}{d} $ means the ratio of $ a $ to $ b $ matches the ratio of $ c $ to $ d $. Proportions help solve real-world problems like scaling recipes, converting units, or analyzing financial ratios.", "## Setting Up the Proportion: $ \frac{7}{4} = \frac{21}{x} $", "To solve $ \frac{7}{4} = \frac{21}{x} $, the first step is recognizing this as a valid proportion where the product of the means equals the product of the extremes—the basis for cross-multiplication. Rearranged, it becomes:\n[\n7x = 4 \ imes 21\n]", "## Cross-Multiplication: Simplify and Solve", "Cross-multiplying means multiplying the numerator of one fraction by the denominator of the other:\n[\n7x = 84\n]", "This step eliminates the fraction and isolates the variable.", "## Solving for $ x $", "Now, divide both sides by 7:\n[\nx = \frac{84}{7} = 12\n]", "Thus, the solution is $ x = 12 $.", "## Verification", "Substituting back into the original proportion:\n[\n\frac{7}{4} = \frac{21}{12}\n]\nSimplify $ \frac{21}{12} $:\n[\n\frac{21 \div 3}{12 \div 3} = \frac{7}{4}\n]\nBoth sides equal $ \frac{7}{4} $, confirming our solution is correct.", "## Why This Matters", "Mastering proportions equips learners with a versatile tool for solving problems involving scaling, rates, and equivalent ratios. Whether calculating medication doses, dividing assets, or comparing data, understanding how to set and solve proportions strengthens logical reasoning and mathematical fluency.", "## Final Answer", "[\n\boxed{12}\n]", "Key Takeaway: Proportions like $ \frac{7}{4} = \frac{21}{x} $ can be solved quickly by cross-multiplying, isolating the variable, and verifying the result—skills vital for academic success and real-world applications.", "---", "Keywords: proportion, solve proportion, cross-multiply, algebraic equation, solve for x, linear equations, math tutorial, proportional reasoning, division of integers, Windows on Math.\nMeta Description: Learn how to solve $ \frac{7}{4} = \frac{21}{x} $ using cross-multiplication. Step-by-step solution with verification, perfect for mastering algebra."]